View source for Physics:Zitterbewegung

{{Short description|Particle effect}}
In [[Physics:Physics|physics]], the '''zitterbewegung''' ({{IPA-de|ˈtsɪtɐ.bəˌveːɡʊŋ}},  {{ety|de|zittern|to tremble, jitter||Bewegung|motion}}) is the theoretical prediction of a rapid oscillatory motion of elementary particles that obey [[Physics:Relativistic wave equations|relativistic wave equations]]. This prediction was first discussed by [[Biography:Gregory Breit|Gregory Breit]] in 1928<ref>{{Cite journal|last=Breit|first=Gregory|date=1928|title=An Interpretation of Dirac's Theory of the Electron|journal=Proceedings of the National Academy of Sciences|language=en|volume=14|issue=7|pages=553–559|doi=10.1073/pnas.14.7.553|issn=0027-8424|pmc=1085609|pmid=16587362|bibcode=1928PNAS...14..553B |doi-access=free}}</ref><ref>{{Cite book|last=Greiner|first=Walter|date=1995|title=Relativistic Quantum Mechanics|url=https://link.springer.com/book/10.1007/978-3-642-88082-7|language=en-gb|doi=10.1007/978-3-642-88082-7|isbn=978-3-540-99535-7|s2cid=124404090 }}</ref> and later by [[Biography:Erwin Schrödinger|Erwin Schrödinger]] in 1930<ref>{{cite book |first= E. |last= Schrödinger |title= Über die kräftefreie Bewegung in der relativistischen Quantenmechanik |language= de |trans-title= On the free movement in relativistic quantum mechanics |pages= 418–428 |date= 1930 |oclc= 881393652 }}</ref><ref>{{cite book |first= E. |last= Schrödinger |title= Zur Quantendynamik des Elektrons |language= de |trans-title= Quantum Dynamics of the Electron |pages= 63–72  |date= 1931}}</ref> as a result of analysis of the [[Physics:Wave packet|wave packet]] solutions of the [[Physics:Dirac equation|Dirac equation]] for relativistic [[Physics:Electron|electron]]s in free space, in which an interference between positive and negative energy states produces an apparent fluctuation (up to the speed of light) of the position of an electron around the median, with an [[Physics:Angular frequency|angular frequency]] of {{math|{{sfrac|2''mc''<sup>2</sup>|''ℏ''}}}}, or approximately {{val|1.6|e=21}} radians per second. 

This apparent oscillatory motion is often interpreted as an artifact of using the Dirac equation in a single particle description and disappears when using [[Physics:Quantum field theory|quantum field theory]]. For the [[Physics:Hydrogen-like atom|hydrogen atom]], the zitterbewegung is related to the [[Physics:Fine structure#Darwin term|Darwin term]], a small correction of the energy level of the [[Physics:Atomic orbital|s-orbitals]].<ref>{{Cite book |last=Tong |first=David |url=https://www.damtp.cam.ac.uk/user/tong/aqm/aqm.pdf |title=Applications of Quantum Mechanics |publisher=University of Cambridge |year=2017}}</ref>

==Theory==

===Free spin-1/2 fermion===
The time-dependent [[Physics:Dirac equation|Dirac equation]] is written as
 
:<math> H \psi (\mathbf{x},t) = i \hbar \frac{\partial\psi}{\partial t} (\mathbf{x},t) </math>,

where <math>\hbar </math> is the [[Physics:Reduced Planck constant|reduced Planck constant]], <math>\psi(\mathbf{x},t) </math> is the [[Wave function|wave function]] ([[Physics:Bispinor|bispinor]]) of a [[Physics:Fermion|fermion]]ic particle spin-½, and {{mvar|H}} is the Dirac [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]] of a [[Physics:Free particle|free particle]]:

:<math> H = \beta mc^2 + \sum_{j = 1}^3 \alpha_j p_j c </math>,

where <math display="inline">m </math> is the mass of the particle, <math display="inline">c</math> is the [[Physics:Speed of light|speed of light]], <math display="inline">p_j </math> is the [[Physics:Momentum operator|momentum operator]], and <math>\beta </math> and <math>\alpha_j </math> are matrices related to the [[Gamma matrices]] <math display="inline">\gamma_\mu </math>, as <math display="inline">\beta=\gamma_0 </math> and <math display="inline"> \alpha_j=\gamma_0\gamma_j </math>.

In the [[Physics:Heisenberg picture|Heisenberg picture]], the time dependence of an arbitrary observable {{mvar|Q}} obeys the equation

:<math> -i \hbar \frac{\partial Q}{\partial t} = \left[ H , Q \right] .</math>

In particular, the time-dependence of the [[Physics:Position operator|position operator]] is given by
:<math>  \frac{\partial x_k(t)}{\partial t} = \frac{i}{\hbar}\left[ H , x_k \right] = c\alpha_k </math>.

where {{math|''x<sub>k</sub>''(''t'')}} is the position operator at time {{mvar|t}}.

The above equation shows that the operator {{mvar|α<sub>k</sub>}} can be interpreted as the {{mvar|k}}-th component of a "velocity operator".

Note that this implies that 

:<math> \left\langle \left(\frac{\partial x_k(t)}{\partial t}\right)^2 \right\rangle=c^2 </math>,

as if the "root mean square speed" in every direction of space is the speed of light.

To add time-dependence to {{mvar|α<sub>k</sub>}}, one implements the Heisenberg picture, which says

:<math> \alpha_k (t) = e^\frac{i H t}{\hbar}\alpha_k e^{-\frac{i H t}{\hbar}}</math>.

The time-dependence of the velocity operator is given by
:<math> \hbar \frac{\partial \alpha_k(t)}{\partial t} = i\left[ H , \alpha_k \right] = 2\left(i \gamma_k m - \sigma_{kl}p^l\right) = 2i\left(p_k-\alpha_kH\right) </math>,

where
:<math>\sigma_{kl} \equiv \frac{i}{2}\left[\gamma_k,\gamma_l\right] .</math>

Now, because both {{mvar|p<sub>k</sub>}} and {{mvar|H}} are time-independent, the above equation can easily be integrated twice to find the explicit time-dependence of the position operator.

First:
:<math>\alpha_k (t) = \left(\alpha_k (0) - c p_k H^{-1}\right) e^{-\frac{2 i H t}{\hbar}} + c p_k H^{-1} </math>,

and finally

:<math> x_k(t) = x_k(0) + c^2 p_k H^{-1} t + \tfrac12 i \hbar c H^{-1} \left( \alpha_k (0) - c p_k H^{-1} \right) \left( e^{-\frac{2 i H t}{\hbar}} - 1 \right) </math>.

The resulting expression consists of an initial position, a motion proportional to time, and an oscillation term with an amplitude equal to the reduced [[Physics:Compton wavelength|Compton wavelength]]. That oscillation term is the so-called zitterbewegung.

==Interpretation==

In quantum mechanics, the zitterbewegung term vanishes on taking expectation values for wave-packets that are made up entirely of positive- (or entirely of negative-) energy waves.  The standard relativistic velocity can be recovered by taking a [[Physics:Foldy–Wouthuysen transformation|Foldy–Wouthuysen transformation]], when the positive and negative components are decoupled.  Thus, we arrive at the interpretation of the zitterbewegung as being caused by interference between positive- and negative-energy wave components.

In [[Physics:Quantum electrodynamics|quantum electrodynamics]] (QED) the negative-energy states are replaced by [[Physics:Positron|positron]] states, and the zitterbewegung is understood as the result of interaction of the electron with spontaneously forming and annihilating electron-positron [[Physics:Pair production|pairs]].<ref>Zhi-Yong, W., & Cai-Dong, X. (2008). Zitterbewegung in quantum field theory. Chinese Physics B, 17(11), 4170.</ref>

More recently,  it has been noted that in the case of free particles it could just be an artifact of the simplified theory. Zitterbewegung appear as due  to the "small components" of the Dirac 4-spinor, due to a little bit of antiparticle mixed up in the particle wavefunction for a nonrelativistic motion. It doesn't appear in the correct [[Physics:Second quantization|second quantized theory]], or rather, it is resolved by using [[Physics:Propagator|Feynman propagators]] and doing [[Physics:Quantum electrodynamics|QED]]. Nevertheless, it is an interesting way to understand certain QED effects heuristically from the single particle picture. <ref>{{Cite web|url=https://physics.stackexchange.com/questions/28672/is-zitterbewegung-an-artefact-of-single-particle-theory|title = Dirac equation - is Zitterbewegung an artefact of single-particle theory?}}</ref>

=== Zigzag picture of fermions ===

An alternative perspective of the physical meaning of zitterbewegung was provided by [[Biography:Roger Penrose|Roger Penrose]],<ref name="PenroseZigzag">{{cite book |last1=Penrose |first1=Roger |title=The Road to Reality |date=2004 |publisher=Alfred A. Knopf |isbn=0-224-04447-8 |edition=Sixth Printing |pages=628–632}}</ref> by observing that the Dirac equation can be reformulated by splitting the four-component [[Physics:Dirac spinor|Dirac spinor]] <math>\psi</math> into a pair of massless [[Physics:Helicity (particle physics)|left-handed and right-handed]] [[Physics:Weyl equation|two-component spinors]] <math>\psi = (\psi_{\rm L}, \psi_{\rm R})</math> (or ''zig'' and ''zag'' components), where each is the source term in the other's equation of motion, with a coupling constant proportional to the original particle's [[Physics:Rest mass|rest mass]] <math>m</math>, as

:<math>
\left\{\begin{matrix}\sigma^\mu \partial_\mu \psi_{\rm R} =  m \psi_{\rm L}\\
\overline{\sigma}^\mu \partial_\mu \psi_{\rm L} =  m \psi_{\rm R}
\end{matrix}\right.
</math>.

The original massive Dirac particle can then be viewed as being composed of two massless components, each of which continually converts itself to the other. Since the components are massless they move at the speed of light, and their spin is constrained to be about the direction of motion, but each has opposite helicity: and since the spin remains constant, the direction of the velocity reverses, leading to the characteristic ''zigzag'' or zitterbewegung motion.

== Experimental simulation ==
Zitterbewegung of a free relativistic particle has never been observed directly, although some authors believe they have found evidence in favor of its existence.<ref>{{Cite journal |last1=Catillon |first1=P. |last2=Cue |first2=N. |last3=Gaillard |first3=M. J. |last4=Genre |first4=R. |last5=Gouanère |first5=M. |last6=Kirsch |first6=R. G. |last7=Poizat |first7=J.-C. |last8=Remillieux |first8=J. |last9=Roussel |first9=L. |last10=Spighel |first10=M. |display-authors=3 |date=2008-07-01 |title=A Search for the de Broglie Particle Internal Clock by Means of Electron Channeling |journal=Foundations of Physics  |volume=38 |issue=7 |pages=659–664 |doi=10.1007/s10701-008-9225-1 |bibcode=2008FoPh...38..659C |s2cid=121875694 |issn=1572-9516}}</ref> It has also been simulated twice in model systems that provide condensed-matter analogues of the relativistic phenomenon. The first example, in 2010, placed a trapped ion in an environment such that the non-relativistic Schrödinger equation for the ion had the same mathematical form as the Dirac equation (although the physical situation is different).<ref>{{cite journal|title=Quantum physics: Trapped ion set to quiver|url=http://www.nature.com/nature/journal/v463/n7277/full/463037a.html|newspaper=Nature News and Views | volume=463|issue=7277|pages=37–39|doi=10.1038/463037a|pmid=20054385|year=2010|last1=Wunderlich|first1=Christof|bibcode=2010Natur.463...37W |doi-access=free}}</ref><ref>{{cite journal|last1=Gerritsma|last2=Kirchmair|last3=Zähringer|last4=Solano|last5=Blatt|last6=Roos|title=Quantum simulation of the Dirac equation|journal=Nature|year=2010|volume=463|issue=7277|doi=10.1038/nature08688|pmid=20054392|arxiv = 0909.0674 |bibcode = 2010Natur.463...68G|pages=68–71|s2cid=4322378}}</ref> Then, in 2013, it was simulated in a setup with [[Physics:Bose–Einstein condensate|Bose–Einstein condensate]]s.<ref>{{cite journal|last1=Leblanc|last2=Beeler|last3=Jimenez-Garcia|last4=Perry|last5=Sugawa|last6=Williams|last7=Spielman|title=Direct observation of zitterbewegung in a Bose–Einstein condensate|journal=[[Physics:New Journal of Physics|New Journal of Physics]]|year=2013|volume=15|issue=7|doi=10.1088/1367-2630/15/7/073011|page=073011|arxiv=1303.0914|s2cid=119190847}}</ref>

Other proposals for condensed-matter analogues include semiconductor nanostructures, [[Physics:Graphene|graphene]] and topological insulators.<ref>{{cite journal|title=Zitterbewegung of Electronic Wave Packets in III-V Zinc-Blende Semiconductor Quantum Wells |last=Schliemann|first=John |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=94 |number=20 |year=2005|pages=206801 |arxiv=cond-mat/0410321|doi=10.1103/PhysRevLett.94.206801 |pmid=16090266|bibcode=2005PhRvL..94t6801S |s2cid=118979437}}</ref><ref>{{cite journal|title=Zitterbewegung, chirality, and minimal conductivity in graphene |last=Katsnelson |first=M. I. |journal=The European Physical Journal B |volume=51 |number=2 |year=2006 |pages=157–160 |arxiv=cond-mat/0512337|doi=10.1140/epjb/e2006-00203-1 |bibcode=2006EPJB...51..157K |s2cid=119353065 }}</ref><ref>{{cite journal|title=Optically engineering the topological properties of a spin Hall insulator |last1=Dóra |first1=Balász |last2=Cayssol |first2=Jérôme |last3=Simon |first3=Ference |last4=Moessner |first4=Roderich |arxiv=1105.5963 |journal=[[Physics:Physical Review Letters|Physical Review Letters]] |volume=108 |number=5 |year=2012 |pages=056602 |doi=10.1103/PhysRevLett.108.056602|pmid=22400947 |bibcode=2012PhRvL.108e6602D |s2cid=15507388 }}</ref><ref>{{cite journal|title=Anomalous Electron Trajectory in Topological Insulators |last1=Shi |first1=Likun |last2=Zhang |first2=Shoucheng |last3=Cheng |first3=Kai |journal=[[Physics:Physical Review B|Physical Review B]] |volume=87 |number=16 |year=2013 |page=161115 |doi=10.1103/PhysRevB.87.161115 |arxiv=1109.4771|bibcode=2013PhRvB..87p1115S |s2cid=118446413 }}</ref>

==See also==
* [[Physics:Casimir effect|Casimir effect]]
* [[Physics:Lamb shift|Lamb shift]]

==References==
{{reflist}}

==Further reading==
* {{cite book |first= A. |last= Messiah |title= Quantum Mechanics |volume= II |chapter= XX, Section 37 |pages= 950–952 |date= 1962 |chapter-url= https://archive.org/details/QuantumMechanicsVolumeIi |chapter-format= pdf |isbn= 9780471597681 }}

== External links ==
* [https://www.newscientist.com/channel/fundamentals/mg19526112.300-the-word-zitterbewebung.html Zitterbewegung in New Scientist]


[[Category:Quantum field theory]]

{{Sourceattribution|Zitterbewegung}}